Problem 56 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) verify the given factor(s) of the function \(f,\) (b) find the remaining factors of \(f,(\mathrm{c})\) use your results to write the complete factorization of \(f,\) and (d) list all real zeros of \(f .\) Confirm your results by using a graphing utility to graph the function. Factor(s) \((x+2)\) \((x+3)\) \((x-5),(x+4)\) \((x+2),(x-4)\) \((2 x+1)\) \((2 x-1)\) Function $$\begin{aligned} f(x)=8 x^{4}-14 x^{3}-71 x^{2} \\ &-10 x+24 \end{aligned}$$

Step-by-Step Solution

Answer

The given factors \(x+2\), \(x+3\) are valid for the provided function \(f(x)\), while \(x-5\), \(x+4\), \(2x+1\) and \(2x-1\) are not. The factorization of \(f(x)\) is \(f(x)=(x+2)^2(x+3)(2x-1)\). The real zeros of the function are x = -2, -2, -3 and 0.5

Step 1: Verify given factors

Substitute each root of given factors into the function \(f(x)\). If \(f(x)\)=0 for that value, then the factor is a factor of the function. So, \(f(-2)\), \(f(-3)\), \(f(5)\), \(f(-4)\), \(f(-2)\), \(f(4)\), \(f(-0.5)\) and \(f(0.5)\) needs to be calculated

Step 2: Find remaining factors

Use synthetic division or polynomial division with verified factors to find the remaining factor(s). Move onto next step if no quotient containing variable remains after division

Step 3: Write the complete factorization

Use the verified and remaining factors to write the complete factorization of the polynomial function.

Step 4: List real zeros

The real zeros of the function are the roots of its factors. List them all as the real zeros of the function.

Step 5: Confirm results using graph

Using a graphing utility, plot the graph of the polynomial function. The x-intercepts of this graph should validate the real zeros that were found in previous step.

Key Concepts

Verifying Factors

To verify a factor of a polynomial, we need to check if it is indeed a true factor. This means when substituted into the polynomial, the factor should make the polynomial equal to zero. In mathematical terms, if \(x = a\) is a root, \(f(a) = 0\). For example, given a factor \(x + 2\), its root would be \(-2\). We substitute \(-2\) into the polynomial function \(f(x)\) and calculate the result. If \(f(-2) = 0\), then \(x + 2\) is a valid factor. Similarly, this process is repeated for each given factor. This step ensures that only the correct factors are used in further polynomial calculations, preventing errors in the subsequent steps of factorization.
It's a crucial validation step, as a mistake here could lead to an incorrect factorization of \(f(x)\).

Synthetic Division

After verifying the factors, we can use synthetic division to find the remaining factors of a polynomial. Synthetic division is a simplified method of dividing polynomials, which involves less writing and fewer operations compared to long division. It is especially useful when dealing with polynomials of higher degrees.

• Set up the synthetic division by writing down the root obtained from the factor, say \(-3\) for \(x + 3\).
• List the coefficients of the polynomial.
• Use these coefficients to perform the synthetic division step by step, bringing down leading terms and multiplying them with the root.

The result will show if the division perfectly ends in zero remainder, helping us identify any additional factors that complete the factorization. This process is repeated until no remainder is left or the polynomial is fully reduced.

Real Zeros

Real zeros of a polynomial are the x-values for which the polynomial equals zero. We can find these zeros by factoring the polynomial completely. Each factor corresponds to a zero of the function; for instance, if \(x - 4\) is a factor, then \(x = 4\) is a real zero.
To list all real zeros, we simply solve for x in each factored term. It's important to remember that the multiplicity of a factor (how many times it appears in the factorization) indicates how many times the corresponding zero will occur for the polynomial. The zeros essentially give us the points at which the graph of the polynomial intersects the x-axis.

Graphing Polynomial

Graphing a polynomial provides a visual representation of the solutions and behavior of the function. By using a graphing utility, like a graphing calculator or software, we plot the polynomial to check for its x-intercepts. These intercepts should match the real zeros calculated through factorization.

• Enter the polynomial equation into the graphing tool.
• Plot the graph and identify where it crosses the x-axis. These crossings, or x-intercepts, are the real zeros of the polynomial.
• Confirm that these x-intercepts correspond with the zeros found through algebraic methods.

Graphing not only confirms the zeros but also shows the general shape and turning points of the polynomial, providing further insight into its properties.